MMW - Chapter 1: Mathematics in our World

Summary

This introductory mathematics lesson with Shara Assassis and Donald in Cape Bayern explores the essence of mathematics as the study of relationships among numbers, quantities, and shapes. It emphasizes math's role in enhancing critical thinking, reasoning, and creativity, and its ability to organize patterns and regularities in the world. The lesson delves into patterns like symmetry, spirals, fractals, and tessellations, introduces the Fibonacci sequence and its connection to the golden ratio, and concludes with the application of mathematics in various real-world phenomena, such as pendulum motion and plane mirror reflection.

Takeaways

• 📚 Mathematics is defined as the study of relationships among numbers, quantities, and shapes, with practical applications like calculating the surface area needed for a cylindrical can cover.
• 🤔 It enhances critical thinking, reasoning, spatial thinking, and creativity by encouraging the search for solutions to mathematical problems, even when the initial approach fails.
• 🔍 Mathematics helps to organize patterns and regularities in the world, which is crucial for understanding natural phenomena and structures.
• 🦋 Symmetry is a pattern where a design or object is identical on both halves, as seen in butterflies and other natural forms.
• 🌀 Spiral patterns are curved shapes that focus on a central point and revolve around it, often found in nature where plants use this form for secure growth.
• 🌿 Fractal patterns are self-replicating shapes that are reduced in size with each repetition, exemplified by the structure of Romanesco broccoli and spider webs.
• 🧩 Tessellations are patterns created by identical shapes fitting together without gaps, such as pineapples and beehives.
• 📈 The Fibonacci sequence, starting with 0, 1, and each subsequent number being the sum of the two preceding ones, is a series that appears in various aspects of nature and mathematics.
• 📅 November 23 is recognized as Fibonacci Day because the date's digits (11/23) correspond to the first four non-zero digits in the Fibonacci sequence.
• 📏 The Golden Ratio, approximately 1.618034, is closely approximated by the ratio of any two successive Fibonacci numbers, but never exactly equal.
• 🔄 Mathematics organizes patterns and regularities such as the motion of a pendulum and the reflection in a plane mirror, providing a mathematical explanation for these phenomena.

Q & A

What is the primary focus of the subject 'Mathematics in the Modern World'?

The primary focus of the subject is to explore the study of relationships among numbers, quantities, and shapes, as well as how mathematics enhances critical thinking, reasoning, spatial thinking, and creativity.

Can you provide an example from the script that illustrates the relationship among numbers and shapes?

An example given in the script is calculating the amount of paper needed to cover a can or cylinder, which requires finding the surface area of the can.

What does the script suggest about the role of mathematics in problem-solving?

The script suggests that mathematics helps in developing critical thinking and finding solutions to problems, emphasizing the persistence in finding the right answer even if the initial approach doesn't work.

What are the different types of patterns discussed in the script?

The script discusses four types of patterns: symmetry, spiral patterns, fractal patterns, and tessellations.

How is the concept of symmetry defined in the script?

In the script, symmetry is defined as a design or pattern that is identical on both halves when folded, using the example of a butterfly with two identical halves.

What is the significance of the Fibonacci sequence in the script?

The Fibonacci sequence is highlighted as a series of numbers where each number is the sum of the two preceding ones, starting from 0 and 1, and it is connected to the golden ratio.

Why is November 23rd referred to as Fibonacci Day?

November 23rd is called Fibonacci Day because the date's digits (11/23) correspond to the first four non-zero digits of the Fibonacci sequence (1, 1, 2, 3).

What is the Golden Ratio and how is it related to the Fibonacci sequence?

The Golden Ratio, denoted by φ (phi), approaches a value of 1.618034. It is related to the Fibonacci sequence because the ratio of any two successive Fibonacci numbers tends to get closer to the Golden Ratio as the numbers get larger.

How does the script connect mathematics to patterns and regularities in the natural world?

The script connects mathematics to natural patterns and regularities by discussing how mathematical concepts can explain phenomena such as the motion of a pendulum and the reflection in a plane mirror.

What is the script's final call to action for the audience?

The script's final call to action is for the audience to reflect on the application of mathematics in their chosen course, encouraging them to consider how mathematical principles are relevant to their field of study.

Outlines

00:00

📘 Introduction to Mathematics and Its Applications

The first paragraph introduces the subject of mathematics in the modern world, with presenters Shara Assassin and Donald In Cape. They begin by defining mathematics as the study of relationships among numbers, quantities, and shapes, using the example of calculating the surface area of a can to determine the amount of paper needed for a cover. The paragraph emphasizes the role of mathematics in enhancing critical thinking, reasoning, spatial thinking, and creativity. It also introduces the concept of patterns in nature and the world, discussing four types of patterns: symmetry, spiral, fractal, and tessellation. Each pattern is exemplified with natural occurrences, such as butterflies for symmetry and pineapples for tessellations. The Fibonacci sequence is introduced, highlighting its significance and the concept that each number in the sequence is the sum of the two preceding ones.

05:02

📐 Fibonacci Sequence and Golden Ratio

The second paragraph delves into the Fibonacci sequence, explaining its significance and the reason why November 23 is celebrated as Fibonacci Day, due to the sequence's first four non-zero digits matching the date. The paragraph explores the relationship between the Fibonacci sequence and the Golden Ratio (phi, approximately 1.618034), noting that the ratio of any two successive Fibonacci numbers approximates the Golden Ratio. Examples are given to illustrate this relationship, such as the ratio of 2 to 3 and 3 to 5, both of which are close to the Golden Ratio. The paragraph concludes with a discussion on the applications of mathematics in understanding patterns and regularities in the world, such as the motion of a pendulum and the reflection in a plane mirror, emphasizing the importance of mathematics in various fields.

Keywords

💡Mathematics

Mathematics is defined as the study of relationships among numbers, quantities, and shapes. It is central to the video's theme as it underpins the discussion on how math is applied in various contexts, such as calculating the surface area of a cylinder to determine the amount of paper needed for a cover. The script uses the example of a can to illustrate the practical application of mathematical concepts.

💡Critical Thinking

Critical thinking is highlighted in the video as a skill enhanced by mathematics. It involves the ability to reason and analyze in order to solve problems effectively. The script mentions that when tackling mathematical problems, individuals are encouraged to think creatively and find multiple solutions, showcasing the importance of critical thinking in mathematical problem-solving.

💡Patterns

Patterns are recurring designs or sequences that can be found in nature and mathematics. The video discusses different types of patterns such as symmetry, spirals, and tessellations, which are organized by mathematical principles. For instance, the script describes a butterfly with two identical halves to illustrate the concept of symmetry.

💡Symmetry

Symmetry refers to a design or pattern that is identical on both halves when folded. In the context of the video, symmetry is presented as one of the natural patterns that can be mathematically described. The script uses the butterfly as an example to demonstrate symmetry, where folding the butterfly would result in two mirror-image halves.

💡Spiral Pattern

A spiral pattern is a curved design that revolves around a central point, often found in nature. The video script explains that plants use spiral forms to grow while maintaining stability. This pattern is an example of how mathematics can describe and understand natural phenomena.

💡Fractal Pattern

Fractal patterns are self-similar shapes that repeat at different scales. The video mentions Romanesco broccoli and spider webs as examples of fractals, illustrating how simple shapes can be repeated to create complex patterns. This concept is integral to understanding the intricate designs found in nature and art.

💡Tessellations

Tessellations are patterns created by identical shapes fitting together without gaps. The video uses pineapple and beehive as examples to explain tessellations. This mathematical concept shows how regularity and repetition can create visually appealing and structurally efficient designs.

💡Fibonacci Sequence

The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, starting from 0 and 1. The video script discusses the sequence as an example of a mathematical pattern found in nature, such as the arrangement of leaves on a stem. The sequence is also related to the Golden Ratio, a concept that appears in various natural and artistic forms.

💡Golden Ratio

The Golden Ratio, denoted by the Greek letter phi (φ), is a mathematical constant approximately equal to 1.618034. The video script explains that the ratio of two successive Fibonacci numbers approaches the Golden Ratio, indicating its prevalence in natural and artistic compositions. This ratio is often considered aesthetically pleasing and is found in various aspects of design and architecture.

💡Regularities

Regularities refer to the consistent patterns or behaviors that can be mathematically described. In the video, regularities are illustrated through examples such as the motion of a pendulum and the reflection in a plane mirror. These phenomena demonstrate how mathematics can explain and predict natural and physical occurrences.

💡Reflection

Reflection, as discussed in the video, pertains to the image produced when light bounces off a surface, such as a mirror. The script uses the law of reflection to explain the regularity in size and distance of the reflected image, showing how mathematical principles can be applied to understand optical phenomena.

Highlights

Mathematics is defined as the study of relationships among numbers, quantities, and shapes.

An example of mathematical application is calculating the surface area needed to cover a cylindrical can.

Mathematics enhances critical thinking, reasoning, spatial thinking, and creativity.

The process of solving mathematical problems involves finding and iterating through various solutions.

Mathematics helps in organizing patterns and regularities in the world.

Types of patterns in nature include symmetry, spiral, fractal, and tessellations.

Symmetrical patterns are designs that are identical on both halves when folded.

Spirals are curved patterns focusing on a central point with circular shapes revolving around it.

Fractal patterns are composed of simple shapes repeated at reduced sizes.

Tessellations are patterns of identical shapes fitting together without gaps.

The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones.

Leonardo Pisano, known as Fibonacci, is credited with the discovery of the Fibonacci sequence.

November 23 is celebrated as Fibonacci Day due to its digits resembling the first four non-zero digits of the sequence.

The Golden Ratio, denoted by φ, is closely related to the Fibonacci sequence.

The ratio of two successive Fibonacci numbers approximates the Golden Ratio but never equals it.

Mathematics organizes the regularities seen in the motion of a pendulum and reflection in a plane mirror.

The lecture concludes with an invitation for students to reflect on the application of mathematics in their chosen courses.

Transcripts

good day welcome to the first lesson of

the subject mathematics in the modern

world

we are shara assassis and donald in cape

bayern to discuss to you the first

chapter let's first define mathematics

for you what is mathematics

math is defined as the study of the

relationships among numbers

quantities and shapes for example

you are to put a cover on a can but for

you to know how much paper you would

need

you have to find first the surface area

of the can or the

cylinder this shows the relationship

among numbers and shapes now

can you think of another example that

shows

relationship among numbers quantities

and shapes

okay another definition of mathematics

is it enhances our critical thinking

skills reasoning

special thinking and creativity

most of us every time we try to answer a

certain mathematical problem

we tend to think of a way to solve the

problem

and if for instance that that certain

way didn't work

we would find another solution again and

we will never stop until we get

to the right answer this is somehow how

mathematics helps us to be a better

thinker and lastly

mathematics helps us organize patterns

and regularities in the world which will

be discussed

on the next slides

okay so we have pattern and numbers in

nature

and the world

so we have uh types of pattern first is

symmetry or the symmetrical pattern it

is a design

or pattern that is identical

on both halves when folded

you may notice that when this butterfly

is folded it will have two identical

halves just like the other examples

presented

okay so that is for symmetrical pattern

next pattern is spiral pattern

which is defined as curved pattern that

focuses on a certain point

and a series of circular shapes that

revolve around it

you may notice that some of the examples

presented

are from the natural environment

the reason why plants use

a uh spiral form

like the plant like this plant presented

is because they are constantly trying to

grow but stay

secure third

is fractal pattern fractal pattern

are built from simple repeated shapes

that are reduced in size

when repeated the two best examples

are romanesco broccoli and the spider

web and last but not the least

desolations

tessellations are created with identical

shapes which

fit together with no gaps so we have

here

pineapple and the beehive

as the example

we will now proceed with the fibonacci

sequence

fibonacci sequence is discovered by

an italian mathematician named leonardo

pisano

his nickname was fibonacci which roughly

means

son of bonacci and

november 23 is the fibonacci day so

later on i will

explain to you why is that so

okay so the sequence goes like this 0

1 1 2 3 5 8 13 21

and so on so what can you notice in the

fibonacci

sequence try to think

what can you notice about this sequence

okay each number in the sequence

is the sum of the two numbers which

preceded

so you may notice that we started from

zero and one so that is the start of the

sequence

and then for us to identify what will be

the next

uh value we would add these two

numbers so we have zero plus one and the

resulting value is the

next number in the sequence which is one

then for us to know to say

the um next term again we would add one

plus one so the answer is two then we

add one plus two the answer is three

we we add two and three the answer is

five

and so on so that is how the sequence

goes and why is november 23 the

fibonacci

day um you may notice that

the digits we have in november 23 is one

one two three and here in the sequence

the first four non-zero digits are one

one two three so that is why

uh i mean which corresponds to the

fibonacci days so that is why

um november 23 is the fibonacci

day okay you can also check this

link to know what is the magic of

fibonacci numbers

next we have the golden

ratio golden ratio is denoted

by fee which approaches a value of

1.618034

okay so next is the relationship

between the fibonacci sequence and the

golden ratio it is said that the ratio

of any two successive fibonacci numbers

is very close to the

golden ratio referred to and represented

as

fee which is i said earlier

approximately equal to

1.618034

okay so where to find the ratio of the

two

successive fibonacci numbers so we would

let a be the smaller number from the

sequence

and you would let b be the larger number

from the

sequence so we have 2 and 3

we will find the ratio b over a which is

equal to 1 to

1.5 which is uh

quite close to the golden ratio

then let's try another consecutive two

consecutive fibonacci numbers

3 and 5. so five divided by three

it's b over eight we have this

um quotient

which is quite equal to the golden ratio

so what can you notice in this uh

table

so you can notice that the bigger the

pair

of the fibonacci numbers considered the

closer the

approximation and you may also ask if

there is a chance that the ratio of the

two successive fibonacci numbers will be

equal to the golden ratio

the answer is no it will just always

be close to the golden ratio but

never equal

next we have the pattern and

regularities in the world as organized

by

mathematics okay the first one

is the motion of a pendulum

the motion of a pendulum shows that the

time

it takes to swing back to its original

position can be

explained by mathematics through

regularities in motion

and the second one is the reflection in

a plane mirror

which shows that the regularity in size

and distance you could see the same size

as the object

in the mirror can be mathematically

explained by

the law of reflection

okay i have your question what do you

think

is the application of mathematics in

your

chosen course okay i want you to reflect

on that

and that ends the first chapter

thank you for listening and i hope that

you learned